What should the range of the output layer be when performing classification?
For this particular classification problem, I would recommend you using a softmax function whose output range is [0,1].
The sum of all outputs should be 1, so an advantage of using a softmax function is that you get a percentage of how confident the network is in this classification.

Side note: As DuttaA has commented, cross entropy loss is a better loss function than the quadratic mean squared error.